Summer Program for Undergraduate Research (SPUR/REU) — Summer 2018

This summer's program has 2 types of projects, SPUR and REU; the difference is the funding available. SPUR is not funded, except for limited funding for Cornell students. (Non-Cornell students are welcome but need to come with their own funding.) REU is funded by the NSF, so support is available for US citizens/residents. See "Support," below, for more information.

PROJECT 1 (SPUR): Analysis on Fractals, directed by Robert Strichartz

Students in this project will study properties of functions defined on fractals. For certain fractals, including the Sierpinski gasket, the Sierpinski carpet, and some of the classical Julia sets, there is now a theory of “differential equations.” (See my book, Differential Equations on Fractals, a tutorial, Princeton University Press, 2006.) One of the goals of this project is to obtain more information about solutions of these fractal differential equations, following up on work that has been done by past REU students. Most of the work on this project will involve both computer experimentation and theoretical study, but individual students may put more emphasis on one or the other. We expect that students will be involved in all stages of the process: planning what examples to study, doing the programming for the computations, and interpreting the results (and attempting to prove the conjectures that come out of the process).

This project will focus on the interplay among combinatorics, discrete geometry, and algebraic topology. Given a convex body in $\mathbb{R}^4$ can it be partitioned into sixteen equal-volume pieces by four affine hyperplanes? Given three red, three green, three blue, and one yellow point in the plane, can these points always be partitioned into four sets with no repeated colors whose convex hulls intersect? These are two of many "simple to state" but open questions in discrete geometry. We will explore a multitude of possible approaches to such intersection and partition results including elementary combinatorial arguments as well as more sophisticated topological methods. The topological methods usually yield continuous generalizations of these convex-geometric results. Recent developments have shown that there are important differences between the affine world of discrete geometry and the continuous world of the corresponding topological generalizations. We will further explore this affine-continuous dichotomy and the role of primes and prime powers in delimiting these two worlds from one another.

Some of these topics could benefit from computer experiments. Prior knowledge of topology is not strictly required. Students who enjoy combinatorial reasoning or want to learn (more) about techniques from algebraic topology are particularly encouraged to apply.

This project will study the "linear algebra" of finite groups. For example, one can define the analogue of a "basis" for any finite group. However, the behavior of these sets can be quite different from bases for vector spaces (e.g., they need not all have the same size, but are related by a Theorem of Tarski). Results from standard linear algebra and the theory of modules are used to suggest questions that should be investigated in the general case. Many such questions have not been previously studied and sometimes offer a new framework to interpret previously isolated results in group theory. One such is the study of certain manipulations of bases which arose in recent years in computational group theory (the product replacement algorithm). Students for this program should have a firm understanding of undergraduate linear algebra and abstract algebra. New topics, which although elementary, are not usually developed in standard undergraduate algebra, will arise naturally here. Students will develop computational tools to study examples using the computer algebra systems GAP and Magma. More information on this project's prerequisites, mechanics and goals can be found in this writeup.

Equations describing optimal behavior often present serious computational challenges. (The quickest driving directions? The most energy-efficient trajectory for a Mars-rover? The risk-of-detection-minimizing flight-plan for a spy plane?) The need for efficient algorithms becomes particularly obvious once you add to the mix the uncertainty about your environment, conflicting goals, and multiple (competing or cooperating) participants. Students participating in this project will investigate the theoretical properties and build fast algorithms for optimal control problems and differential games. Successful candidates will need good programming skills, previous exposure to ordinary differential equations and numerical computing. Some background in the following areas will also be helpful, but is not expected or required: partial differential equations, design and analysis of algorithms, probability theory. More on this project: https://pi.math.cornell.edu/~vlad/reu

SUPPORT: The REU project (project 4) is supported by the NSF, so students who are U.S. citizens or permanent residents are eligible to apply for funding (\$5,000). Non-Cornell students interested in SPUR projects (1, 2, or 3) will have to come with their own funding; Cornell students may apply for local support of \$2,500. Participants will arrange for their own housing; we will assist with local contact information.

HOW TO APPLY: Please submit an application (through the SPUR web site) that includes a statement about your background, educational goals, and your scientific interests. Include whatever further information you consider relevant. (Be sure to include information about your computer experience.) You will be able to upload letters of recommendation and transcripts (unofficial transcripts are accepted) via the SPUR website. Two letters of recommendation are required. Uploading your letters and transcripts is preferred and is the most efficient means of processing. The reference writer is notified to upload their letter once you enter their information on the application.

DEADLINE: February 22, 2018. All materials must be received by this date. Late applications will not be accepted. You will receive notification some time in March. We will be adhering to the uniform first reply date for REU programs, so no student will be asked to reply to an acceptance offer before March 8.

If you have comments, questions, or concerns, please send e-mail to the SPUR coordinators at mathreu@cornell.edu.